3. EXPERIMENTAL TECHNIQUES
3.9. Instrument simulation and optimization
The modelling of neutron optics components as well as complete neutron scattering instruments, mainly by the Monte Carlo (MC) ray tracing method, has been developed intensively over the past decade. It has become indispensable for any project building a new instrument. Rapidly growing demand for MC modelling comes particularly from large neutron scattering centres with their ambitious instrument development programmes. The Millennium Programme of the ILL in Grenoble, new instrumentation for the FRM II reactor in Munich, the
second target station of ISIS and the ESS project are the best known examples.
This effort, which results in up to an order of magnitude more efficient use of otherwise limited neutron sources, opens many new experimental possibilities in various fields of science, particularly in condensed matter physics, materials research, macromolecular chemistry and microbiology. Such progress in neutron instrumentation is possible thanks to the introduction of new concepts in the application of neutron optics, such as focusing reflective and refractive optics, adaptive multicrystal Bragg mirrors, fast position sensitive neutron detectors, etc. In this respect, MC simulations proved to be very useful for the quantitative estimation of future instrument properties and in the optimization of the design of neutron optical systems in order to achieve a maximum possible gain in intensity, resolution or both. The higher neutron fluxes achieved by focusing neutron optics often lead to a more complicated non-Gaussian instrumental response compared with older instruments employing tightly collimated beams. As a result, data analysis methods developed earlier for non-focusing instruments with Gaussian response become inadequate. Here, MC simulations are also useful, as they yield a more realistic description of instrumental response function, which can be taken into account when analysing experimental data.
Tools for realistic MC simulations of neutron experiments have been developed. At present, development of the method of virtual experiments is in progress, which will be useful for instrument building, experiment planning and advanced data analysis. Simulations should become an everyday tool for instrument builders, instrument scientists and even for user groups. Reliability in simulation results will be ensured by extensive tests and comparisons with experiments.
While the development of neutron ray tracing software began in the 1970s, it has been significantly accelerated by the advent of powerful desktop computers in the 1990s and led to the creation of several software packages capable of the realistic modelling of modern neutron optics devices. They include the programs NISP [55], IDEAS [56], McStas [57], VITESS [58] and RESTRAX [59]. Of course, MC simulations always involve a trade-off between the level of physical reality implemented in the description of neutron transport and computing speed.
Consequently, these programs differ in both the physical models underlying the simulation of particular components and the structure of their code, depending on the different purposes they have been written for. While some put emphasis on modularity (McStas, VITESS, NISP), which permits easy the incorporation of new components and testing of new ideas of experimental techniques, others, like RESTRAX, trade part of their flexibility for a highly efficient sampling strategy.
Thanks to this approach, RESTRAX is significantly faster in many situations and hence more suitable for applications requiring a large amount of simulated data, such as optimization of one instrument configuration and analysis of experimental
data. As such, RESTRAX has proved to be useful in a growing number of instrument development projects, including those carried out by leading edge neutron facilities such as the ILL in Grenoble, HZB, FRM II [60–66]. Unlike the other programs, RESTRAX provides integrated tools for analysing experimental data from two- and three axis neutron scattering instruments, with more realistic (simulated) description of instrumental response than can be achieved with analogous analytical methods. It is thus useful not only for instrument scientists, but also for neutron users, who need to plan their experiments, treat experimental data more accurately or to resolve problems related to instrumental artefacts.
Recently, SIMRES has been upgraded to allow for simulations of a much larger variety of instrument configurations. During the past decade, SIMRES has been checked several times against other ray tracing packages (McStas, VITESS, NISP) and has proved to be reliable, providing identical results when simulating identical configurations. The main advantages of using SIMRES in simulating crystal diffractometers are (a) the ability to describe diffraction on bent and mosaic crystals realistically and (b) high speed, which permits the scanning of wide ranges of multiple variable instrument parameters. Nevertheless, as with all other simulation packages, the depth of physical reality implemented in the software is limited.
In relation to the CRP project, very successful MC simulations have been performed for optimization of parameters of stress diffractometers in HZB [66], South African Nuclear Energy Corporation (Necsa) in South Africa, Korea Atomic Energy Research Institute (KAERI) in the Republic of Korea [67]
and China Institute of Atomic Energy (CIAE) in China [68]. The following paragraphs summarize the approximations adopted in description of individual diffractometer components used in this work:
— Information on a neutron source can be given in various ways. The most accurate one is a lookup table, which may include nearly arbitrary energy, spatial and angular flux distribution. The source is usually assumed to be homogeneous and isotropic with Maxwellian flux distribution.
— All collimators and slits are assumed to perfectly absorb, i.e. no trajectories through the walls or missing the collimator opening are processed.
— For the crystal monochromator, elastically bent perfect crystals are assumed. For neutron trajectories, the bent crystals are deterministic and approximated by straight lines with the turn points, which are determined from Bragg’s law, calculated for a uniformly (cylindrically) bent perfect crystal. The events are weighted by diffraction probability derived from dynamical diffraction theory. This model is in very good agreement with experiments and with theory for the curvatures used on monochromators (1/RH >~ 0.001 m−1). Reflectivity is calculated from table values of
scattering amplitudes and includes the Debye-Waller factor. Absorption is calculated on the basis of the semi-empirical model developed by Freund [69]. This model describes beam attenuation due to neutron capture, incoherent scattering, single and multiphonon scattering. It provides energy and temperature dependent absorption coefficients in good agreement with experimental data on single crystals, within the energy range of thermal neutrons.
— For simulations, a simplified model of a polycrystalline α-Fe sample is usually assumed, neglecting texture, microstrains, grain size effects and the Debye-Waller factor.
— A position sensitive detector with 100% efficiency is assumed. The spatial resolution is described by Gaussian response with a required variance e.g. 1 mm (in both directions).
— In the simulation program, all parameters related to the instrument configuration are included (e.g. the cross-section of the beam, the dimensions of the horizontally and vertically focusing monochromator, monochromator take-off angle, the scattering angle on the sample, the monochromator–sample distance, the sample–detector distance, etc.).
The vertical curvature of the monochromator, the crystal thickness and, in the case of a sandwich, the number of wafers, are optimized during the simulation process.
The simulation process is sequential, which means that any cross-talk between different components is excluded. In practice, cross-talk effects are mediated by parasitic or incoherent scattering, which are anyway not processed by the program. Note that cross-talk between segments of a focusing crystal has to be taken into account since such an array is treated as a single component. As a result of the MC simulation, resolution and projections of 3-D intensity profiles of the beam of ‘useful’ neutrons between the monochromator and the sample as well as between the sample and the detector can be received [66–68].
For users and instrument scientists without experience in such calculations, it is recommended to first contact a scientist with ample experience in MC simulations in order to clarify necessary questions and requirements.
3.9.2. Matrix simulation technique
Another alternative of simulation of the experiment on a stress diffractometer is provided by the so called matrix technique developed by a Romanian group [69–71]. It appears to be a convenient procedure to compute the resolution and luminosity properties, to perform the configuration optimization, to compute the beam size or any properties defining an experimental configuration, as the normal
approximation is still acceptable. However, for more sophisticated purposes, when a correct description of the line profile is needed, a MC procedure should be used. The computation procedure of the matrix technique involves several steps (for details see Refs [70–72]). The first one is to choose the initial variables of the problem, which define the neutron trajectories between the source and the detector. For a crystal neutron diffractometer, the initial variables, which define the vector R, are all the relevant spatial coordinates plus the variable of the reflectivity curve of the monochromator. The normal approximation of the probability distribution of the initial variables can be constructed with a known transmission matrix, S. This amounts to replacing the actual shape of the spectrometer elements with Gaussian distributions with the same second order moments. It is assumed that the neutrons are scattered with a probability given by the real geometrical shapes of the spectrometer elements; this approximation can be improved by defining probability distributions taking into account extinction and absorption.
The next step is to account for the presence of soller collimators, neutron guides, coarse collimators and slits (a slit is defined as a coarse collimator with zero length). The neutron guide is assumed to be a soller collimator with a wave length dependent angular divergence given by the total reflex critical angle; this approximates the actual rectangular transmission function of the guide with a triangular one, with the same second order moments.
The computational procedure then serves as a base for a computer program (DAX program) to evaluate the resolution and intensity properties of a given experimental configuration or to optimize the parameters defining such a configuration. The DAX program was realized some years ago [73] and since then it has been considerably improved to become a really powerful instrument for designing and optimizing experimental settings. It was intensively used in designing the focusing configurations in Bucharest and Piteşti, Romania, and at the MURR reactor in the USA for optimizing the already existing configuration according to experimental requirements.
The DAX program allows for obtaining the resolution function, line widths and intensities as well as the optimum values for some relevant parameters. The program can be used with configurations with a monochromator group of one or two crystals, conventionally with plane crystals but for focusing, curved crystals are used; with or without soller collimators (the former is the conventional set-up;
the latter the focusing set-up) and with or without diaphragms or neutron guides.
The crystals can be mosaic or perfect; an option for vibrating crystals is also included. The index in the program that corresponds to the experimental set-up is selected according to the ‘how to use’ instructions.
To reduce the involved matrix dimensions, the horizontal and vertical plane computations are separated at first and combined at the end. This is possible
owing to the lack of correlation between the corresponding variables. The experimental configuration optimization can be performed either analytically using the corresponding optimization conditions or numerically, by minimizing the optimization parameter w1/22/Imax (w1/2 is the full line width and Imax is the maximum peak intensity). For Poisson statistics, a minimum value of this ratio assures the optimum conditions to separate two partially overlapping lines. For numerical optimization, the analytical conditions, when they really exist, are the zeroth order approximation. When the analytical condition does not exist, the input value for the numerically optimization process is chosen arbitrarily.
For analytical optimization, the general focusing conditions [73] provide the optimum monochromator radius of curvature, sample monochromator distance and the optimum sample orientation. There are 12 parameters (14 for two crystal monochromators) for which numerical optimization can be performed, the radius (ii) of curvature, the cutting angle, the mosaic spread, the crystal thickness, the sample orientation or thickness, the soller collimators’ divergences (if soller collimators are used as in the case of conventional configurations), the monochromator–sample and sample–detector distances and the detector window width. Though all 12 (or 14) parameters can be optimized together, this is not desirable because the computing time would be too long; for reasonable computing time the number of parameters to be optimized at the same time should not exceed six. A library with the relevant data concerning the most frequently used crystals is included in the program. If the crystal does not belong to this list, the corresponding data must be input by the user.